On improving a one-layer ocean model with thermodynamics

Abstract

A popular method used to incorporate thermodynamic processes in a shallow water model (e.g. one used to study the upper layer of the ocean) is to allow for density variations in time and horizontal position, but keep all dynamical fields as depth independent. This is achieved by replacing the horizontal pressure gradient by its vertical average. These models have limitations, for instance they cannot represent the ‘thermal wind’ balance (between the horizontal density gradient and the vertical shear of the velocity) which dominates at low frequencies. A new model is now proposed which uses velocity and density fields varying linearly with depth, with coefficients that are functions of horizontal position and time. This model can explicitly represent the thermal wind balance, but its use is not restricted to low- frequency dynamics.Volume, mass, buoyancy variance, energy and momentum are conserved in the new model. Furthermore, these integrals of motion have the same dependence on the dynamical fields as the exact (continuously stratified) case. The evolution of the three components of the absolute vorticity field are correctly represented. Conservation of density–potential vorticity is not fulfilled, though, owing to artificial removal of the vertical curvature of the velocity field.The integrals of motion are used to construct a ‘free energy’ [Escr ]f, which is quadratic to the lowest order in the deviation from a steady state with (at most) a uniform velocity field. [Escr ]fis positive definite, and therefore the free evolution of the system cannot lead to an ‘explosion’ of the dynamical fields. (This is not the case if the velocity shear and/or the density vertical gradient is excluded in the model, which results in a non-negative definite free energy.)In a model with one active layer, linear waves on top of a steady state with no currents are, to a very good approximation, those of the first two vertical modes of the continuously stratified model. These are the familiar geophysical gravity and vortical waves (e.g. Poincaré, Rossby, and coastal Kelvin waves at mid-latitudes, equatorial waves, etc.).Finally, baroclinic instability is well represented in the new model. For long perturbations (wavelengths of the order of the deformation radius of the first mode) the agreement with more precise calculations is excellent. On the other hand, the comparison with the eigenvalues of Eady's problem (which corresponds to wavelengths of the order of the deformation radius of the second mode) shows differences of the order of 40%. Nevertheless, the new model does have a high-wavenumber cutoff, even though it is constrained to linear profiles in depth and therefore cannot reproduce the exponential trapping of Eady's problem eigensolutions.In sum, the integrals of motion, vorticity dynamics, free waves and baroclinic instability results all give confidence in the new model. Its main novelty, however, lies in the ability to incorporate thermodynamic processes.